As a consequence, \(\sum _{n \le x} \alpha _y(n) = \sum _{d \text{ is }y\text{-friable}} \mu (d) \lfloor x/d \rfloor \)  and this may be used to obtain an error term involving friable numbers [1]}:
\(\sum _{n \le x} \alpha _y(n) = xP_y + O(\Psi (x,y)).\) 
